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For one relation to be isotonic BUT NOT the dual you would need a way to separate them. Isotonic is a word I've read though. But "isotonic: monotonic but where the relations are visually facing the same way" is not how I want to define it! Will look into later Alec (talk) 08:44, 9 April 2016 (UTC)

Oops, I fail to get your hint. Quite unclear, what do you mean? Boris (talk) 09:01, 9 April 2016 (UTC)
Another option: "order preserving" versus "order inverting". Not sure whether it is in use. Boris (talk) 09:06, 9 April 2016 (UTC)
While I get what you mean by "order preserving" and "order reversing" I cannot come up with a definition. Suppose we have:
  • Isotonic: [ilmath]\forall a,b\in X[a\mathcal{R} b\implies f(a)\mathcal{S}f(b)][/ilmath]
This only works if [ilmath]f[/ilmath] is "order preserving" itself. Suppose [ilmath]\mathcal{R} [/ilmath] and [ilmath]\mathcal{S} [/ilmath] are [ilmath]\le[/ilmath] and [ilmath]f:\mathbb{R} \rightarrow\mathbb{R} [/ilmath], if we define [ilmath]f:x\mapsto -x[/ilmath] this is no longer isotonic.
BUT! If we define [ilmath]\mathcal{S} [/ilmath] as [ilmath]\ge[/ilmath] it is now "isotonic".
If both [ilmath]\mathcal{ R } [/ilmath] and [ilmath]\mathcal{ S } [/ilmath] are "to the right" (eg [ilmath]\le[/ilmath]) this works as expected, as if they're both to the left (eg [ilmath]\ge[/ilmath]) then it's actually the same thing. (But I cannot define "to the left" formally, this is what I will investigate.)
  • That is: [ilmath]\forall a,b\in X[a\le b\implies f(a)\le f(b)]\leftrightarrow\forall a,b\in X[a\ge b\implies f(a)\ge f(b)][/ilmath] where [ilmath]\ge[/ilmath] is the dual of whatever [ilmath]\le[/ilmath] is.
However I cannot define "to the left" (as these are dual concepts, I don't expect to be able to UNLESS there is some "natural order preserving map", [ilmath]f[/ilmath], then the above definition works)
Do you see what I mean? Alec (talk) 10:33, 9 April 2016 (UTC)
PS: We can however say "[ilmath]f[/ilmath] is isotonic with respect to (two partial orders)" what I'm saying here is we first need to determine if both relations "are facing the same direction") Alec (talk) 10:47, 9 April 2016 (UTC)
Now I see. But I think, these "directions" (left and right) exist only on paper (or screen etc), but not in the mathematical universe. Notations are a matter of metamathematics. In mathematics, we are given two partially ordered sets; that is all. Some maps preserve order, some invert. But, yes, we may ask, what happens if we endow the same set with the opposite order (thus constructing another partially ordered set). Boris (talk) 11:26, 9 April 2016 (UTC)
Indeed. I cannot think of a way to distinguish [ilmath]\le[/ilmath] from [ilmath]\ge[/ilmath] (on say [ilmath]\mathbb{R} [/ilmath] without a notion of "think of [ilmath]>[/ilmath] like a crocodile, it wants to eat the bigger number (eg [ilmath]5>4[/ilmath])" (this is how we are taught to remember [ilmath]>[/ilmath] and [ilmath]<[/ilmath] in school by the way) and of course as they are duals....
This is what I'll look into. I have heard the word "isotonic" before, but I wont make a definition without references as I am not confident (where as I know monotonic. I am confident enough to leave finding a reference for later). I shall add/work-towards [ilmath]\sigma[/ilmath]-ideal today. Alec (talk) 11:45, 9 April 2016 (UTC)
Screenshot of weird definition
Okay. I've been reviewing some things in preparation for creating long-missing pages (like supremum) and I had an order-theory book I got from the library website. It has a few errors (mainly typos, borderline misleading sentences....) and I thought I'd look up monotone. This is what I got: (please look at the picture linked on the right)
According to this isotone and monotone (monotonic) are the same things. That's interesting. But it goes further than that! It defines "antitone". Not "formally" either. It relies on the "human knowledge" of "is the symbol we use to represent the order going the right way"
Too much terminology is based on "monotonic" (like sequences, monotonically increasing/decreasing) and I like the idea of "isotonic" (increasing) for "same way" and "antitonic" for the contra way (which will be decreasing).
I do not feel that these terms have (or even ought) to be used, as "monotonically increasing", "strictly monotonically increasing", ditto for decreasing are something that first-years encounter with sequences.
I justify "increasing" meaning "non-decreasing" as we speak of [ilmath]\ge[/ilmath] by default, and use strictly ([ilmath]>[/ilmath]) when we need to specify. So monotonically increasing means "[ilmath]\le[/ilmath]" on both sides.
  • To sum up, I see no need to mention isotone and antitone beyond a note about foreign terminology used.
Not a huge issue, but do you have any thoughts? Oh! I also don't like "antitone", if anything it should be contratone as it goes "against" rather than being the same. I will not (try to) start a new convention though, especially when I'm ready to dismiss anything beyond monotonic. Alec (talk) 21:18, 9 April 2016 (UTC)